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See detailMixing Taylor shifts and universal Taylor series
Beise, Peter; Meyrath, Thierry UL; Müller, Jürgen

in Bulletin of the London Mathematical Society (2015), 47

It is known that, generically, Taylor series of functions holomorphic in a simply connected complex domain exhibit maximal erratic behaviour outside the domain (so-called universality) and overconvergence ... [more ▼]

It is known that, generically, Taylor series of functions holomorphic in a simply connected complex domain exhibit maximal erratic behaviour outside the domain (so-called universality) and overconvergence in parts of the domain. Our aim is to show how the theory of universal Taylor series can be put into the framework of linear dynamics. This leads to a unified approach to universality and overconvergence and yields new insight into the boundary behaviour of Taylor series. [less ▲]

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See detailNew identities for small hyperbolic surfaces
Hu, Hengnan UL; Tan, Ser Peow

in Bulletin of the London Mathematical Society (2014), 46(5), 10211031

Luo and Tan gave a new identity for hyperbolic surfaces with/without geodesic boundary in terms of dilogarithms of the lengths of simple closed geodesics on embedded three-holed spheres or one-holed tori ... [more ▼]

Luo and Tan gave a new identity for hyperbolic surfaces with/without geodesic boundary in terms of dilogarithms of the lengths of simple closed geodesics on embedded three-holed spheres or one-holed tori in Luo and Tan [‘A dilogarithm identity on moduli spaces of curves’, J. Differential Geom., Preprint, 2011, arXiv:1102.2133[math.GT]]. However, the identity was trivial for a hyperbolic one-holed torus with geodesic boundary. In this paper, we adapt the argument from Luo and Tan to give an identity for hyperbolic tori with one geodesic boundary or cusp in terms of dilogarithm functions on the set of lengths of simple closed geodesics on the torus. As a corollary, we are also able to express the Luo–Tan identity as a sum over all immersed three-holed spheres P which are embeddings when restricted to the interior of P [less ▲]

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